3.44.94 \(\int \frac {1-2 e^7+(7-6 e^7) \log (x)+6 \log ^2(x)}{(x-2 e^7 x) \log (x)+2 x \log ^2(x)} \, dx\)

Optimal. Leaf size=22 \[ 2+\log \left (3 x^3 \log (x) \left (1+2 \left (-e^7+\log (x)\right )\right )\right ) \]

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Rubi [A]  time = 0.05, antiderivative size = 20, normalized size of antiderivative = 0.91, number of steps used = 4, number of rules used = 2, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.044, Rules used = {6, 893} \begin {gather*} 3 \log (x)+\log (\log (x))+\log \left (2 \log (x)-2 e^7+1\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - 2*E^7 + (7 - 6*E^7)*Log[x] + 6*Log[x]^2)/((x - 2*E^7*x)*Log[x] + 2*x*Log[x]^2),x]

[Out]

3*Log[x] + Log[Log[x]] + Log[1 - 2*E^7 + 2*Log[x]]

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 893

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &
& NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && I
ntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-\operatorname {Subst}\left (\int \frac {-1+2 e^7-7 x+6 e^7 x-6 x^2}{x \left (1-2 e^7+2 x\right )} \, dx,x,\log (x)\right )\\ &=-\operatorname {Subst}\left (\int \frac {-1+2 e^7+\left (-7+6 e^7\right ) x-6 x^2}{x \left (1-2 e^7+2 x\right )} \, dx,x,\log (x)\right )\\ &=-\operatorname {Subst}\left (\int \left (-3+\frac {2}{-1+2 e^7-2 x}-\frac {1}{x}\right ) \, dx,x,\log (x)\right )\\ &=3 \log (x)+\log (\log (x))+\log \left (1-2 e^7+2 \log (x)\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.11, size = 20, normalized size = 0.91 \begin {gather*} 3 \log (x)+\log (\log (x))+\log \left (1-2 e^7+2 \log (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 - 2*E^7 + (7 - 6*E^7)*Log[x] + 6*Log[x]^2)/((x - 2*E^7*x)*Log[x] + 2*x*Log[x]^2),x]

[Out]

3*Log[x] + Log[Log[x]] + Log[1 - 2*E^7 + 2*Log[x]]

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fricas [A]  time = 1.38, size = 19, normalized size = 0.86 \begin {gather*} 3 \, \log \relax (x) + \log \left (-2 \, e^{7} + 2 \, \log \relax (x) + 1\right ) + \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*log(x)^2+(-6*exp(7)+7)*log(x)-2*exp(7)+1)/(2*x*log(x)^2+(-2*x*exp(7)+x)*log(x)),x, algorithm="fri
cas")

[Out]

3*log(x) + log(-2*e^7 + 2*log(x) + 1) + log(log(x))

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giac [A]  time = 0.22, size = 36, normalized size = 1.64 \begin {gather*} \frac {1}{2} \, \log \left (\pi ^{2} {\left (\mathrm {sgn}\relax (x) - 1\right )}^{2} + {\left (2 \, e^{7} - 2 \, \log \left ({\left | x \right |}\right ) - 1\right )}^{2}\right ) + 3 \, \log \relax (x) + \log \left ({\left | \log \relax (x) \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*log(x)^2+(-6*exp(7)+7)*log(x)-2*exp(7)+1)/(2*x*log(x)^2+(-2*x*exp(7)+x)*log(x)),x, algorithm="gia
c")

[Out]

1/2*log(pi^2*(sgn(x) - 1)^2 + (2*e^7 - 2*log(abs(x)) - 1)^2) + 3*log(x) + log(abs(log(x)))

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maple [A]  time = 0.06, size = 20, normalized size = 0.91

method	result	size
default	\(3 \ln \relax (x )+\ln \left (1+2 \ln \relax (x )-2 \,{\mathrm e}^{7}\right )+\ln \left (\ln \relax (x )\right )\)	\(20\)
norman	\(3 \ln \relax (x )+\ln \left (\ln \relax (x )\right )+\ln \left (2 \,{\mathrm e}^{7}-2 \ln \relax (x )-1\right )\)	\(20\)
risch	\(3 \ln \relax (x )+\ln \left (\ln \relax (x )^{2}+\left (-{\mathrm e}^{7}+\frac {1}{2}\right ) \ln \relax (x )\right )\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((6*ln(x)^2+(-6*exp(7)+7)*ln(x)-2*exp(7)+1)/(2*x*ln(x)^2+(-2*x*exp(7)+x)*ln(x)),x,method=_RETURNVERBOSE)

[Out]

3*ln(x)+ln(1+2*ln(x)-2*exp(7))+ln(ln(x))

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maxima [B]  time = 0.46, size = 373, normalized size = 16.95 \begin {gather*} -6 \, {\left (\frac {\log \left (-e^{7} + \log \relax (x) + \frac {1}{2}\right )}{2 \, e^{7} - 1} - \frac {\log \left (\log \relax (x)\right )}{2 \, e^{7} - 1}\right )} e^{7} \log \relax (x) + 6 \, {\left (\frac {\log \left (-e^{7} + \log \relax (x) + \frac {1}{2}\right )}{2 \, e^{7} - 1} - \frac {\log \left (\log \relax (x)\right )}{2 \, e^{7} - 1}\right )} \log \relax (x)^{2} - 3 \, {\left (\frac {{\left (2 \, e^{7} - 2 \, \log \relax (x) - 1\right )} \log \left (-e^{7} + \log \relax (x) + \frac {1}{2}\right ) - 2 \, e^{7} + 2 \, \log \relax (x) + 1}{2 \, e^{7} - 1} + \frac {2 \, {\left (\log \relax (x) \log \left (\log \relax (x)\right ) - \log \relax (x)\right )}}{2 \, e^{7} - 1}\right )} e^{7} - 2 \, {\left (\frac {\log \left (-e^{7} + \log \relax (x) + \frac {1}{2}\right )}{2 \, e^{7} - 1} - \frac {\log \left (\log \relax (x)\right )}{2 \, e^{7} - 1}\right )} e^{7} + 7 \, {\left (\frac {\log \left (-e^{7} + \log \relax (x) + \frac {1}{2}\right )}{2 \, e^{7} - 1} - \frac {\log \left (\log \relax (x)\right )}{2 \, e^{7} - 1}\right )} \log \relax (x) + \frac {3 \, {\left (4 \, \log \relax (2) \log \relax (x)^{2} + 4 \, \log \relax (x)^{2} \log \left (\log \relax (x)\right ) + 2 \, {\left (2 \, e^{7} - 1\right )} \log \relax (x) - {\left (4 \, \log \relax (x)^{2} - 4 \, e^{14} + 4 \, e^{7} - 1\right )} \log \left (-2 \, e^{7} + 2 \, \log \relax (x) + 1\right )\right )}}{2 \, {\left (2 \, e^{7} - 1\right )}} + \frac {7 \, {\left ({\left (2 \, e^{7} - 2 \, \log \relax (x) - 1\right )} \log \left (-e^{7} + \log \relax (x) + \frac {1}{2}\right ) - 2 \, e^{7} + 2 \, \log \relax (x) + 1\right )}}{2 \, {\left (2 \, e^{7} - 1\right )}} + \frac {7 \, {\left (\log \relax (x) \log \left (\log \relax (x)\right ) - \log \relax (x)\right )}}{2 \, e^{7} - 1} + \frac {\log \left (-e^{7} + \log \relax (x) + \frac {1}{2}\right )}{2 \, e^{7} - 1} - \frac {\log \left (\log \relax (x)\right )}{2 \, e^{7} - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*log(x)^2+(-6*exp(7)+7)*log(x)-2*exp(7)+1)/(2*x*log(x)^2+(-2*x*exp(7)+x)*log(x)),x, algorithm="max
ima")

[Out]

-6*(log(-e^7 + log(x) + 1/2)/(2*e^7 - 1) - log(log(x))/(2*e^7 - 1))*e^7*log(x) + 6*(log(-e^7 + log(x) + 1/2)/(
2*e^7 - 1) - log(log(x))/(2*e^7 - 1))*log(x)^2 - 3*(((2*e^7 - 2*log(x) - 1)*log(-e^7 + log(x) + 1/2) - 2*e^7 +
2*log(x) + 1)/(2*e^7 - 1) + 2*(log(x)*log(log(x)) - log(x))/(2*e^7 - 1))*e^7 - 2*(log(-e^7 + log(x) + 1/2)/(2
*e^7 - 1) - log(log(x))/(2*e^7 - 1))*e^7 + 7*(log(-e^7 + log(x) + 1/2)/(2*e^7 - 1) - log(log(x))/(2*e^7 - 1))*
log(x) + 3/2*(4*log(2)*log(x)^2 + 4*log(x)^2*log(log(x)) + 2*(2*e^7 - 1)*log(x) - (4*log(x)^2 - 4*e^14 + 4*e^7
- 1)*log(-2*e^7 + 2*log(x) + 1))/(2*e^7 - 1) + 7/2*((2*e^7 - 2*log(x) - 1)*log(-e^7 + log(x) + 1/2) - 2*e^7 +
2*log(x) + 1)/(2*e^7 - 1) + 7*(log(x)*log(log(x)) - log(x))/(2*e^7 - 1) + log(-e^7 + log(x) + 1/2)/(2*e^7 - 1
) - log(log(x))/(2*e^7 - 1)

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mupad [B]  time = 3.16, size = 19, normalized size = 0.86 \begin {gather*} \ln \left (\ln \relax (x)\right )+\ln \left (2\,\ln \relax (x)-2\,{\mathrm {e}}^7+1\right )+3\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*exp(7) - 6*log(x)^2 + log(x)*(6*exp(7) - 7) - 1)/(2*x*log(x)^2 + log(x)*(x - 2*x*exp(7))),x)

[Out]

log(log(x)) + log(2*log(x) - 2*exp(7) + 1) + 3*log(x)

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sympy [A]  time = 0.17, size = 20, normalized size = 0.91 \begin {gather*} 3 \log {\relax (x )} + \log {\left (\log {\relax (x )}^{2} + \left (\frac {1}{2} - e^{7}\right ) \log {\relax (x )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*ln(x)**2+(-6*exp(7)+7)*ln(x)-2*exp(7)+1)/(2*x*ln(x)**2+(-2*x*exp(7)+x)*ln(x)),x)

[Out]

3*log(x) + log(log(x)**2 + (1/2 - exp(7))*log(x))

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